Finite Group Databasebeta

← #30600157

Subgroups — #30600157

Order 15616.

610 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840778C211yesmaximal, characteristic
2#21461673C211yesmaximal, characteristic
3C2×Q8⋊#92420C212yesmaximal, complemented
4#21460655C212yesmaximal, complemented
5C22⋊Q8×Dic61C211yesmaximal, characteristic
6#16840778—24no—
7C2×C4⋊#92420—21no—
8C2×C4⋊#92420C2212yes—
9C2×C22⋊#92420—21no—
10C2×C22⋊#92420—21no—
11C2×C4×#92420C2212yes—
12#16840628—21no—
13Q8⋊#92420—24no—
14C2×Q8×Dic61C2211yescharacteristic
15C22×C4×Dic61C2211yescharacteristic
16#16839669C411yescharacteristic
17#16839661C411yescharacteristic
18#16839648—21no—
19#16839637—21no—
20#16839622C411yescharacteristic
21C4⋊C4×Dic61—21no—
22C4⋊C4×Dic61C2211yescharacteristic
23C22⋊C4×Dic61—21no—
24C2×C22⋊Q8×C61C411yescharacteristic
25C22×#92420—22no—
26C22×#92420—22no—
27C4⋊#92420—24no—
28C4⋊#92420—22no—
29C4⋊#92420—22no—
30C22⋊#92420—24no—
31C22⋊#92420—42no—
32C4×#92420—22no—
33C4×#92420D414yes—
34C23×Dic61D411yescharacteristic
35C23×Dic61D411yescharacteristic
36Q8×Dic61—22no—
37Q8×Dic61—22no—
38C2×C4⋊Dic61—21no—
39C2×C4⋊Dic61—21no—
40C2×C4⋊Dic61C2×C411yescharacteristic
41C2×C4⋊Dic61C2×C411yescharacteristic
42C2×C22⋊Dic61—21no—
43C2×C4×Dic61—22no—
44C2×C4×Dic61—22no—
45C2×C4×Dic61—21no—
46C2×C4×Dic61—21no—
47C2×C4×Dic61C2×C411yescharacteristic
48C2×C4×Dic61D411yescharacteristic
49C2×C4×Dic61D411yescharacteristic
50C2×C4×Dic61C2311yescharacteristic
51#14519684—21no—
52#14519684—21no—
53#14519684—21no—
54#14519684—21no—
55#14519684—21no—
56C22×Q8×C61C2×C411yescharacteristic
57C23×C4×C61C2×C411yescharacteristic
58C22⋊Q8×C61—22no—
59C22⋊Q8×C61C814yescomplemented
60C2×C4⋊C4×C61—21no—
61C2×C4⋊C4×C61C2×C411yescharacteristic
62C2×C22⋊C4×C61—21no—
63C24×C61C22⋊C411yescharacteristic
64C2×Q8×C61—21no—
65C2×Q8×C61—22no—
66C2×Q8×C61—22no—
67C2×Q8×C61C2×C812yes—
68C22×C4×C61—22no—
69C22×C4×C61—22no—
70C22×C4×C61—21no—
71C22×C4×C61—21no—
72C22×C4×C61C22⋊C411yescharacteristic
73C22×C4×C61C2×C812yes—
74C22×C4×C61M1612yes—
75C22×C4×C61C22×C411yescharacteristic
76C22×Dic61—21no—
77C22×Dic61—22no—
78C22×Dic61—21no—
79C22×Dic61—21no—
80C22×Dic61—21no—
81C22×Dic61—22no—
82C22×Dic61—21no—
83C22×Dic61—21no—
84C22×Dic61—41no—
85C22×Dic61—41no—
86C22×Dic61C22⋊C411yescharacteristic
87C22×Dic61C22⋊C411yescharacteristic
88C22×Dic61C2×D411yescharacteristic
89C22×Dic61C2×D411yescharacteristic
90C2×#92420—24no—
91C2×#92420—42no—
92C2×#92420—42no—
93C2×#92420—42no—
94C2×#92420—42no—
95C2×#92420—42no—
96C4⋊C4×C61—24no—
97C4⋊C4×C61—21no—
98C4⋊C4×C61C2×C812yes—
99C22⋊C4×C61—24no—
100C22⋊Dic61—42no—
101C4⋊Dic61—22no—
102C4⋊Dic61—22no—
103C4⋊Dic61—42no—
104C4⋊Dic61—41no—
105C4⋊Dic61—41no—
106C4×Dic61—22no—
107C4×Dic61—21no—
108C4×Dic61—22no—
109C4×Dic61—21no—
110C4×Dic61—42no—
111C4×Dic61—41no—
112C4×Dic61—41no—
113C4×Dic61C2×D412yes—
114C4×Dic61#4112yes—
115C23×C61—21no—
116C23×C61—21no—
117C23×C61—21no—
118C23×C61—21no—
119C23×C61—21no—
120C23×C61—21no—
121C23×C61C22⋊C812yes—
122C23×C61C2×C22⋊C411yescharacteristic
123Q8×C61—24no—
124Q8×C61—42no—
125C2×C4×C61—21no—
126C2×C4×C61—21no—
127C2×C4×C61—24no—
128C2×C4×C61—24no—
129C2×C4×C61—22no—
130C2×C4×C61—22no—
131C2×C4×C61—41no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61—41no—
135C2×C4×C61C22⋊C812yes—
136C2×C4×C61C4×D412yes—
137C2×C4×C61C22×C811yescharacteristic
138C2×C4×C61C2×M1611yescharacteristic
139C2×C4×C61#13112yes—
140C2×Dic61—21no—
141C2×Dic61—21no—
142C2×Dic61—21no—
143C2×Dic61—21no—
144C2×Dic61—21no—
145C2×Dic61—21no—
146C2×Dic61—21no—
147C2×Dic61—21no—
148C2×Dic61—42no—
149C2×Dic61—42no—
150C2×Dic61—42no—
151C2×Dic61—42no—
152C2×Dic61—41no—
153C2×Dic61—41no—
154C2×Dic61—41no—
155C2×Dic61—41no—
156C2×Dic61—41no—
157C2×Dic61—41no—
158C2×Dic61#12011yescharacteristic
159C2×Dic61#12112yes—
160C2×Dic61#12311yescharacteristic
161C2×Dic61AGL(1,ℤ/8)11yescharacteristic
162C2×Dic61#13712yes—
163C2×Dic61#13711yescharacteristic
164#92420—44no—
165#92420—82no—
166#9073—611nomaximal, Sylow
167C22×C61—22no—
168C22×C61—22no—
169C22×C61—22no—
170C22×C61—41no—
171C22×C61—41no—
172C22×C61—41no—
173C22×C61—41no—
174C22×C61#38611yescharacteristic
175C22×C61C2×C22⋊C811yescharacteristic
176C22×C61#40811yescharacteristic
177C22×C61#41711yescharacteristic
178C22×C61#41911yescharacteristic
179C22×C61#42111yescharacteristic
180C22×C61#42111yescharacteristic
181C4×C61—22no—
182C4×C61—42no—
183C4×C61—42no—
184C4×C61C8×D412yes—
185C4×C61#43512yes—
186Dic61—21no—
187Dic61—21no—
188Dic61—41no—
189Dic61—41no—
190Dic61—81no—
191Dic61—81no—
192Dic61#46812yes—
193Dic61#47012yes—
194C2×#431—611no—
195C4×C22⋊Q8—611no—
196#1797—611no—
197C2×Q8⋊C8—612no—
198#1279—612no—
199C2×C61—42no—
200C2×C61#147711yescharacteristic
201C2×C61#147911yescharacteristic
202C2×C61#149511yescharacteristic
203C2×C61#149711yescharacteristic
204C2×C61#186911yescharacteristic
205C2×C61#189411yescharacteristic
206C2×C61#189711yescharacteristic
207C2×C22⋊Q8#692311yescharacteristic
208C2×C4×Q8—611no—
209C22×C42—611no—
210#431—1224no—
211C2×C4⋊C8—612no—
212C2×C4⋊C8—1221no—
213C2×C22⋊C8—1221no—
214C2×C22⋊C8—1221no—
215C2×C4×C8—612no—
216#391—611no—
217#389—611no—
218#387—1221no—
219#385—1221no—
220#380—611no—
221C4×C4⋊C4—611no—
222C4×C4⋊C4—1221no—
223C4×C22⋊C4—1221no—
224#336—1221no—
225Q8⋊C8—1224no—
226C61#907311yescharacteristic, complemented, Sylow
227C22×Q8C2×#692311yescharacteristic
228C23×C4—611no—
229C23×C4—611no—
230C23×C4C2×#692311yescharacteristic
231C22×C8—1222no—
232C22×C8—1222no—
233C22⋊Q8—22no—
234C22⋊Q8#9242014yescomplemented
235C4×Q8—1222no—
236C4×Q8—1222no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—611no—
239C2×C4⋊C4—611no—
240C2×C4⋊C4—1221no—
241C2×C4⋊C4—1221no—
242C2×C4⋊C4C2×#692311yescharacteristic
243C2×C22⋊C4—21no—
244C2×C22⋊C4—1221no—
245C2×C42—611no—
246C2×C42—611no—
247C2×C42—611no—
248C2×C42—611no—
249C2×C42—1222no—
250C2×C42—1222no—
251C2×C42—1221no—
252C2×C42—1221no—
253C4⋊C8—1222no—
254C4⋊C8—1222no—
255C4⋊C8—1224no—
256C22⋊C8—1224no—
257C22⋊C8—2442no—
258C4×C8—614no—
259C4×C8—1222no—
260#95—1221no—
261#95—1221no—
262#95—1221no—
263#95—1221no—
264#95—1221no—
265C24C22⋊#692311yescharacteristic
266C2×Q8—21no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8C2×#9242012yes—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—611no—
275C22×C4—611no—
276C22×C4—611no—
277C22×C4—611no—
278C22×C4—1221no—
279C22×C4—1221no—
280C22×C4—1222no—
281C22×C4—1221no—
282C22×C4—1221no—
283C22×C4—1222no—
284C22×C4—1221no—
285C22×C4—1221no—
286C22×C4—2441no—
287C22×C4—2441no—
288C22×C4C2×#9242012yes—
289C22×C4#1175821712yes—
290C22×C4C22⋊#692311yescharacteristic
291C22×C4C22×#692311yescharacteristic
292C2×C8—1224no—
293C2×C8—2442no—
294C2×C8—2442no—
295C2×C8—2442no—
296C2×C8—2442no—
297C2×C8—2442no—
298C4⋊C4—24no—
299C4⋊C4—21no—
300C4⋊C4—1222no—
301C4⋊C4—1222no—
302C4⋊C4—2441no—
303C4⋊C4—2441no—
304C4⋊C4—2442no—
305C4⋊C4C2×#9242012yes—
306C22⋊C4—24no—
307C22⋊C4—2442no—
308C42—612no—
309C42—612no—
310C42—1221no—
311C42—1222no—
312C42—1221no—
313C42—1222no—
314C42—2441no—
315C42—2441no—
316C42—2442no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23C22⋊#9242012yes—
324C23C2×C22⋊#692311yescharacteristic
325Q8—24no—
326Q8—42no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—612no—
338C2×C4—611no—
339C2×C4—611no—
340C2×C4—612no—
341C2×C4—611no—
342C2×C4—611no—
343C2×C4—1221no—
344C2×C4—1221no—
345C2×C4—1221no—
346C2×C4—1221no—
347C2×C4—1221no—
348C2×C4—1221no—
349C2×C4—1221no—
350C2×C4—1221no—
351C2×C4—2442no—
352C2×C4—2442no—
353C2×C4—2442no—
354C2×C4—2442no—
355C2×C4—2441no—
356C2×C4—2441no—
357C2×C4—2441no—
358C2×C4—2441no—
359C2×C4—2441no—
360C2×C4—2441no—
361C2×C4C22⋊#9242012yes—
362C2×C4D4×#692312yes—
363C2×C4C22×#9242011yescharacteristic
364C2×C4C2×#1175821711yescharacteristic
365C2×C4#1451986412yes—
366C8—4882no—
367C8—2444no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22#1684071011yescharacteristic
376C22C2×C22⋊#9242011yescharacteristic
377C22#1684073511yescharacteristic
378C22#1684075311yescharacteristic
379C22#1684075511yescharacteristic
380C22#1684075911yescharacteristic
381C22#1684076111yescharacteristic
382C4—22no—
383C4—42no—
384C4—42no—
385C4—612no—
386C4—612no—
387C4—1221no—
388C4—1221no—
389C4—4881no—
390C4—4881no—
391C4—2441no—
392C4—2441no—
393C4D4×#9242012yes—
394C4#1684079312yes—
395C2—42no—
396C2#2146111411yescharacteristic
397C2#2146112111yescharacteristic
398C2#2146117611yescharacteristic
399C2#2146118111yescharacteristic
400C2#2146179211yescharacteristic
401C2#2146183411yescharacteristic
402C2#2146183811yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC2×C4×C61C22×C8
FrattiniC22×C4C22×#6923
FittingC2×C22⋊Q8×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC24×C61C22⋊C4
cosocleC2×C4×Dic61C23
hypercenterC2×C22⋊Q8#6923
hypocenterC61#9073
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C2×Q8⋊#92420C2semidirect
4. #21460655C2semidirect
59. C22⋊Q8×C61C8semidirect
226. C61#9073semidirect
234. C22⋊Q8#92420semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168407782
#214616732
C2×Q8⋊#924202
#214606552
C22⋊Q8×Dic612
#907361